Some papers filling in the gaps of my integral arguments are found here. Note they are more geared towards analytic number theory, and more general asymptotic constructions.
You can find that my analytic continuation of \(F\) is a naive idea, compared to what's presented here (which includes my result):
http://pcmap.unizar.es/~chelo/investig/p...sality.pdf
You can also see more generally, that this is related to Mellin convolution (which I didn't bother touching on, because I'm sure it would confuse everyone. I Mellin convoluted to pull out the Borel sum). Also note, this paper even approaches the Analytic Number theory \(\approx\) Quantum physics realm. I apologize, but that's much more my mathematical history.
https://www.sciencedirect.com/science/ar...759500002E
You can download the full pdf to this source from there too. This is more related to Exponential sums and signal theory, but it covers the basics which I tried to speedrun in the beginning of my post. But this helps see some of the strong asymptotics you can get from Mellin transforms. You'll notice if you read to about 10-15 pages in that we start to discuss expansions near \(0\), and how we can better explain the mellin transform.
Also, we have used the amazing book https://www.fing.edu.uy/~cerminar/Complex_Analysis.pdf by Stein & Shakarchi; specifically their description of bounds on Fourier transforms in the complex plane. Which through a variable change, gave us our \(|F(x+iy)| < M e^{-\frac{\pi}{2}|y|}\).
I'll try to find a specific paper that covered much more of the work towards the end of my result, but was phrased in a more restrictive manner. It was a statement that:
\[
F(z-N) \sim c^N N!\\
\]
So the worst this object can grow as \(\Re(z) \to -\infty\) is factorially (minus an exponential); which would mean \(\frac{1}{\Gamma(1-z)}\) would provide the appropriate decay to ensure our Contour integral produced a convergent series.
You can find that my analytic continuation of \(F\) is a naive idea, compared to what's presented here (which includes my result):
http://pcmap.unizar.es/~chelo/investig/p...sality.pdf
You can also see more generally, that this is related to Mellin convolution (which I didn't bother touching on, because I'm sure it would confuse everyone. I Mellin convoluted to pull out the Borel sum). Also note, this paper even approaches the Analytic Number theory \(\approx\) Quantum physics realm. I apologize, but that's much more my mathematical history.
https://www.sciencedirect.com/science/ar...759500002E
You can download the full pdf to this source from there too. This is more related to Exponential sums and signal theory, but it covers the basics which I tried to speedrun in the beginning of my post. But this helps see some of the strong asymptotics you can get from Mellin transforms. You'll notice if you read to about 10-15 pages in that we start to discuss expansions near \(0\), and how we can better explain the mellin transform.
Also, we have used the amazing book https://www.fing.edu.uy/~cerminar/Complex_Analysis.pdf by Stein & Shakarchi; specifically their description of bounds on Fourier transforms in the complex plane. Which through a variable change, gave us our \(|F(x+iy)| < M e^{-\frac{\pi}{2}|y|}\).
I'll try to find a specific paper that covered much more of the work towards the end of my result, but was phrased in a more restrictive manner. It was a statement that:
\[
F(z-N) \sim c^N N!\\
\]
So the worst this object can grow as \(\Re(z) \to -\infty\) is factorially (minus an exponential); which would mean \(\frac{1}{\Gamma(1-z)}\) would provide the appropriate decay to ensure our Contour integral produced a convergent series.